Cheg 258 final project

Do not discuss this problem with others!

Due in the main office no later than 10AM on the
day the final exam would have been held (Tuesday,
5/3/94).  The problem -must- be received by this
time, or I must have a written medical excuse.

Plug flow reactor with recycle:

  We consider the behavior of a plug flow reactor
with a recycle loop.  In this reactor we have two
chemicals in the feed, A and B.  These undergo the
reaction A + B => C which is desirable, and the 
reaction A + C => D which is undesirable.  We want
to maximize the yield (production per unit feed) of
C.  The catalyst used in the reaction is inhibited by
the reactant A, so that at both very high and very
low concentrations of A the reaction is slow.  It is
proposed that a plug flow reactor of length L be
modified with a recycle stream to improve the yield.
We have the system:
             _________________________
feed        |                         |     product
************|   plug flow reactor     |************
      *     |_________________________|    *
      *                                    *
      **************************************
                     recycle stream

  The flow rate of the feed is Qf, the flow rate of
the recycle is a*Qf, so the flow through the reactor
itself is (1+a)*Qf.  If the cross-sectional area of
the reactor is area, then the velocity is given by
u=(1+a)*Qf/area.  We let the rate of the first reaction
(per unit volume of reactor) be given by R1 and the
rate of the second by R2.  The consumption of A is
thus given by:

u*(dCa/dz) = -R1 - R2

and the consumption of B is given by:

u*(dCb/dz) = -R1

The feed concentrations of A and B are Caf and Cbf,
respectively.  The concentrations of C and D can be
determined from stoichiometry to be:

Cc = 2*(Cbf - Cb) - (Caf - Ca)

and

Cd = (Caf - Ca) - (Cbf - Cb)

  The reaction kinetics are given by:

R1 = k1*Ca*Cb/(1 + lambda*Ca^2)

and

R2 = k2*Ca*Cc

where k1, k2, and lambda are constants.  The inlet
concentration (z=0) is a combination of the feed
concentration and the recycle concentration.  The
latter is just the concentration at z=L.  Thus:

Ca0 = (Caf + a*CaL)/(1 + a)

and

Cb0 = (Cbf + a*CbL)/(1 + a)

  Using all of this information, we want to determine
the yield of C as a function of the degree of recycle
a.  Solve the following problems:

1).  For the condition a=0 calculate the yield of C
and the selectivity (the ratio of C to D).  This is
just an initial value problem.  Plot the concentration
of all species as a function of position.

2).  For the recycle condition a=1 repeat the calculation
of the yield and the selectivity.  This involves using
the shooting method.  Plot the concentration of all
species as a function of position.

3).  Plot the yield and the selectivity as a function
of the recycle parameter rv=a/(a+1) which has the useful
property of mapping a onto the interval [0,1].  Determine
where the yield and selectivity are at a maximum (these
will be for different values of a).  You can determine
the location of these maxima graphically, but use an
automatic algorithm to get the selectivity and the
yield for each value of rv or a.


Use the parameters:

Caf = 1.8
Cbf = 1
Qf = 1
area = 1
L = 1
lambda = 10 
k1 = 50 
k2 = 5